Problem: Simplify and expand the following expression: $ \dfrac{2}{5k + 8}-\dfrac{2k}{k - 6} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5k + 8)(k - 6)$ Multiply the first term by $\dfrac{k - 6}{k - 6}$ $ \begin{align*} \dfrac{2}{5k + 8} \times \dfrac{k - 6}{k - 6} & = \dfrac{(2)(k - 6)}{(5k + 8)(k - 6)} \\ & = \dfrac{2k - 12}{(5k + 8)(k - 6)}\end{align*} $ Multiply the second term by $\dfrac{5k + 8}{5k + 8}$ $ \begin{align*} \dfrac{2k}{k - 6} \times \dfrac{5k + 8}{5k + 8} & = \dfrac{(2k)(5k + 8)}{(k - 6)(5k + 8)} \\ & = \dfrac{10k^2 + 16k}{(k - 6)(5k + 8)}\end{align*} $ Now we have: $ = \dfrac{2k - 12}{(5k + 8)(k - 6)} - \dfrac{10k^2 + 16k}{(k - 6)(5k + 8)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2k - 12 - (10k^2 + 16k)}{(5k + 8)(k - 6)} $ $ = \dfrac{2k - 12 - 10k^2 - 16k}{(5k + 8)(k - 6)} $ $ = \dfrac{-14k - 12 - 10k^2}{(5k + 8)(k - 6)}$ Expand the denominator: $ = \dfrac{-14k - 12 - 10k^2}{5k^2 - 22k - 48}$